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This lecture covers the process of rotating an object around an arbitrary axis in 3D space, structured in several transformation steps. Key concepts include translating the axis to the origin, rotating around the X-axis to the XZ plane, and further manipulating the axis along the Y-axis before performing the final rotation. The method also includes inverse transformations to restore the original axis. Each step utilizes matrix transformations for clear visualization and understanding. Ideal for students in electrical and computer engineering.
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Lecture 3 Supplement Prof. Hsien-Hsin Sean Lee School of Electrical and Computer Engineering Georgia Institute of Technology
Rotation Along an Arbitrary Axis • Transform the axis to cross origin (0,0,0) • Transform the axis around X onto the XZ plane • Transform the axis around Y onto the Z axis • Perform Rotation (angle=) • Inverse Transform of 3. • Inverse Transform of 2. • Inverse Transform of 1.
y x z Step 1: Translate to (0, 0, 0) • For easier viewing, I use RHS. Rotating axis
y x z Step 2: Rotate around X, project to XZ y (a, b, c) b z x (into the screen) c • Let’s pick up a random point (a, b, c) on the axis to figure out cos
Step 3: Transform around Y onto the Z axis y a x y (into the screen) x (a’, 0, c’) z z
